We thank Dan Abramovich, Simon Donaldson, Alessandro Ghigi, Hiroshi Iritani, Johan de Jong, Dmitri Panov, Miles Reid, Yann Rollin, James Sparks, Balázs Szendrői and Dror Varolin for useful conversations. Abramovich and Brendan Hassett have also recently studied moduli of orbifolds and weighted projective embeddings [1]}, though from a very different and much more professional point of view. In particular they do not use GIT and are mainly interested in the singularities that occur in the compactification; here we are only concerned with smooth orbifolds for the link to differential geometry. JR received support from NSF Grant DMS-0700419 and Marie Curie Grant PIRG-GA-2008-230920 and RT held a Royal Society University Research Fellowship while this work was carried out.
